Review of methods for fitting time-series models with process and observation error and likelihood calculations for nonlinear, non-Gaussian state-space models

A key challenge for analyzing fisheries time-series data has been to incorporate sources of uncertainty such as process error, observation error, and model-structure uncertainty. Recent years have seen promising advances in methods for handling the first two together in a state-space framework, but likelihood calculations for state-space models require high-dimensional integrals, which make their use computationally challenging. The first section of this paper reviews model-fitting methods that use a state-space model structure, including errors-in-variables methods, Bayesian methods that do and do not use the state-space likelihood, and the possibility of classical likelihood analysis with nonlinear, non-Gaussian state-space models. It also discusses the relationship between true likelihood calculations and errors-in-variables likelihoods, as well as the role of Monte Carlo methods in implementing Bayesian and/or state-space model analyses. The second section introduces a numerical method for calculating state-space likelihoods without Monte Carlo methods and gives examples in a classical maximum-likelihood framework. The method is applicable when the dimension of the state space at each time step is low. Although recent advances in model-fitting and analysis methods are promising, inferences from noisy data and complex processes will continue to be variable and uncertain.